Laplace Decomposition Method to Study Solitary Wave Solutions of Coupled Nonlinear Partial Differential Equation

Abstract

Analytical and numerical solutions are obtained for coupled nonlinear partial differential equation by the well-known Laplace decomposition method. We combined Laplace transform and Adomain decomposition method and present a new approach for solving coupled Schrödinger-Korteweg-de Vries (Sch-KdV) equation. The method does not need linearization, weak nonlinearity assumptions, or perturbation theory. We compared the numerical solutions with corresponding analytical solutions.

1. Introduction

Systems of partial differential equations have attracted much attention in a variety of applied sciences because of their wide applicability. These systems were formally derived to describe wave propagation to model the shallow water waves [1–5] and to examine some chemical reaction-diffusion model of Brusselator [4–6]. Some of the commonly used methods to solve these equations are the method of characteristics, the Riemann invariants, and Adomian decomposition method [6].

In this work, we used Laplace decomposition method introduced by Khuri [7, 8] which is further used by Yusufoglu to solve Duffing equation [9] and Elgasery for Falkner-Skan equations [10]. This technique, modified by Hussain and Khan [11], illustrates how the Laplace transform may be used to approximate the solutions of the nonlinear partial differential equations by extending the decomposition method [12, 13].

2. Laplace Decomposition Method (LDM)

In this section, we outline the main steps of the method. We consider the nonlinear partial differential equations in an operator form:𝐿𝐿_𝑡𝑡𝑢+𝑅₁(𝑢,𝑣)+𝑁₁(𝑢,𝑣)=𝑓₁,𝑣+𝑅₂(𝑢,𝑣)+𝑁₂(𝑢,𝑣)=𝑓₂.(1) With initial data 𝑣𝑢((𝑥𝑥,,00))==𝑔𝑔₁₂(𝑥),(𝑥),(2) where 𝐿_𝑡=𝜕/𝜕𝑡 is a first-order partial differential operator, 𝑅₁, 𝑅₂ and 𝑁₁, 𝑁₂ are linear and nonlinear operators, respectively, and 𝑓₁,𝑓₂ are the source terms. By applying the Laplace transform to both sides of (1) and using initial conditions (2), we have 𝐿𝐿𝐿𝐿_𝑡𝑡𝑢+𝐿𝑅₁(𝑢,𝑣)+𝐿𝑁₁(𝑢,𝑣)=𝐿𝑓₁,(𝑣)+𝐿𝑅₂(𝑢,𝑣)+𝐿𝑁₂(𝑢,𝑣)=𝐿𝑓₂.(3) By the differentiation property of Laplace transform, we get 𝐿(𝑢)=𝑔¹(𝑥)𝑝+𝐿𝑓¹𝑝−1𝑝𝐿𝑅¹𝐿(𝑣)=𝑔₂(𝑢,𝑣)+𝐿𝑁₁(𝑢,𝑣),(𝑥)𝑝+𝐿𝑓²𝑝−1𝑝𝐿𝑅²(𝑢,𝑣)+𝐿𝑁₂(𝑢,𝑣),(4) where “𝑝”   is Laplace domain function. Solutions 𝑢(𝑥,𝑡) and 𝑣(𝑥,𝑡) in LDM are defined as 𝑢(𝑥,𝑡)=^∞_𝑛=0𝑢_𝑛,𝑣(𝑥,𝑡)=^∞_𝑛=0𝑣_𝑛.(5) The nonlinear terms 𝑁₁,𝑁₂, represented by infinite series, 𝑁₁(𝑥,𝑡)=^∞_𝑛=0𝐴_𝑛,𝑁₂(𝑥,𝑡)=^∞_𝑛=0𝐵_𝑛(6) are the Adomian polynomials [14], generated for all forms of nonlinearity and they are determined by the following relations: 𝐴_𝑛=1𝑛!𝑑𝑑𝜆^𝑛𝑛𝑁^1∞_𝑖=0𝜆^𝑖𝑢_𝑖_𝜆=0𝐵_𝑛,=1𝑛!𝑑𝑑𝜆^𝑛𝑛𝑁^2∞_𝑖=0𝜆^𝑖𝑣_𝑖_𝜆=0.(7) Substituting (5) and (6) into (4), gives 𝐿^∞_𝑛=0𝑢_𝑛=𝑔¹(𝑥)𝑝+𝐿𝑓¹−1𝑝𝑝𝐿𝑅¹^∞_𝑛=0𝑢_𝑛,^∞_𝑛=0𝑣_𝑛+𝐿_∞_𝑛=0𝐴_𝑛𝐿_∞,_𝑛=0𝑣_𝑛=𝑔²(𝑥)𝑝+𝐿𝑓²−1𝑝𝑝𝐿𝑅²^∞_𝑛=0𝑢_𝑛,^∞_𝑛=0𝑣_𝑛+𝐿_∞_𝑛=0𝐵_𝑛.(8) Applying the linearity of the Laplace transform, we deduct the following recursive relations: 𝐿𝑢₀=𝑔¹(𝑥)𝑝+𝐿𝑓¹𝐿𝑣₀𝑝,=𝑔²(𝑥)𝑝+𝐿𝑓²𝐿𝑢₁𝑝,𝐿𝑣₁=−1𝑝𝐿𝑅¹𝑢_0,𝑣₀−1𝑝𝐿𝐴⁰,=−1𝑝𝐿𝑅²𝑢_0,𝑣₀−1𝑝𝐿𝐵⁰.(9) In general, for 𝑘≥1, the recursive relations are given by 𝐿𝑢_𝑘+1=−1𝑝𝐿𝑅¹𝑢_𝑘,𝑣_𝑘−1𝐿𝑣_𝑘+1𝑝𝐿𝐴^𝑘,=−1𝑝𝐿𝑅²𝑢_𝑘,𝑣_𝑘−1𝑝𝐿𝐵^𝑘.(10) Applying the inverse Laplace transform, we can evaluate 𝑢_𝑘 and 𝑣_𝑘 (𝑘≥0). In some cases the exact solution in the closed form can be obtained.

3. Application

At the classical level, a set of coupled nonlinear wave equations describes the interaction between high-frequency and low-frequency waves [15], and the calculation of exact and numerical solutions of the equations, in particular, travelling wave solutions, plays an important role in wave-wave interaction and soliton theory [1, 16].

We consider the Schrödinger-KdV (Sch-KdV) equation as a model for the interaction of long and short nonlinear waves: 𝜂𝑖𝐸_𝑡=𝐸_𝑥𝑥+𝐸𝜂,_𝑡=−6𝐸𝜂_𝑥−𝜂_𝑥𝑥𝑥+||𝐸||²_𝑥.(11) With initial conditions 𝐸(𝜂𝑥(,𝑥0,)0=)=2√8𝑘2𝑘²₂1−3tanh²(𝑘𝑥)𝑒^𝑖𝛼𝑥,−𝛼3−6𝑘²tanh²(𝑘𝑥),(12) where 𝛼, 𝑘 are arbitrary constant.

With Laplace decomposition method on (11) and using the differentiation property of Laplace transform, initial conditions and the inverse Laplace transforms are^∞_𝑛=0𝐸_𝑛(𝑥,𝑡)=𝐸(𝑥,0)−𝑖𝐿⁻¹1𝑝𝐿𝐸^𝑛𝑥𝑥+^∞_𝑛=0𝐴_𝑛(𝜂,𝐸)_∞,_𝑛=0=𝜂𝜂_𝑛(𝑥(𝑥,,0𝑡))+𝐿₋₁1𝑝𝐿^∞_𝑛=0𝐵_𝑛(𝐸)−6^∞_𝑛=0𝐶_𝑛(𝜂,𝐸)−𝜂_{𝑛𝑥𝑥𝑥},(13) where (𝑥,𝑡)=∑^∞𝑛=0𝐸_𝑛,𝜂(𝑥,𝑡)=∑^∞𝑛=0𝜂_𝑛, and ∑^∞𝑛=0𝐴_𝑛(𝜂,𝐸)=𝜂𝐸   ∑^∞𝑛=0𝐵_𝑛(𝐸)=(|𝐸|²)_𝑥   ∑^∞𝑛=0𝐶_𝑛(𝜂,𝐸)=𝐸𝜂_𝑥 are adomian polynomials that represent nonlinear terms.

So the recursive relation is deduced as 𝐸₁𝐸(𝑥,0)=𝐸₀=2√2𝑘²1−3tanh²(𝑘𝑥)𝑒^𝑖𝛼𝑥,(𝑥,𝑡)=−𝑖𝐿⁻¹1𝑝𝐿𝐸^0𝑥𝑥+^∞_𝑛=0𝐸𝐴₀(𝜂,𝐸)_𝑛+1,(𝑥,𝑡)=−𝑖𝐿⁻¹1𝑝𝐿𝐸^𝑛𝑥𝑥+^∞_𝑛=0𝜂(𝑥,0)=𝜂₀𝐴_𝑛(𝜂,𝐸),=8𝑘²−𝛼3−6𝑘²𝜂tanh²(𝑘𝑥),₁(𝑥,𝑡)=𝐿⁻¹1𝑝𝐿^∞_𝑛=0𝐵₀(𝐸)−6^∞_𝑛=0𝐶₀(𝜂,𝐸)−𝜂_{0𝑥𝑥𝑥}𝜂_𝑛+1,(𝑥,𝑡)=𝐿⁻¹1𝑝𝐿^∞_𝑛=0𝐵_𝑛(𝐸)−6^∞_𝑛=0𝐶_𝑛(𝜂,𝐸)−𝜂_{𝑛𝑥𝑥𝑥}𝑛≥1.,(14) By this recursive relation we can find other components of the solution as 𝐸₁(𝑥,𝑡)=−𝑖𝐿⁻¹1𝑝𝐿𝐸^0𝑥𝑥𝐸₁+𝜂₀𝐸₀,(𝑥,𝑡)=2√2𝑘²𝑡×33𝛼₂×cos+(𝛼𝛼+𝑥)1+0𝑘3²6𝑘(𝛼1s−ec3h₂tanh(𝑘𝑥))×tanh(𝑘𝑥)cos(𝛼𝑥)+(𝑖k2x√)2𝑘²𝑡×3𝛼₂3×−3(16−+𝑘𝛼3𝛼s+teacnh1h₂0𝑘(𝑘²𝑥))sin(𝛼𝑥)𝜂(𝑘𝑥)tanh(𝑘𝑥)cos(𝛼𝑥)₁,(𝑥,𝑡)=𝐿⁻¹1𝑝𝐿||𝐸⁰||²𝜂₁_𝑥−6𝐸₀𝜂_0x−𝜂_{0𝑥𝑥𝑥},(𝑥,𝑡)=−+4184𝑘4√⁵𝑡sech²(𝑘𝑥)4−9tanh²(𝑘𝑥)×1−32𝑘t⁵a𝑡nshe₂ch²(𝑘𝑥)tanh(𝑘𝑥)×sech₂(𝑘𝑥)cos(𝛼𝑥)+144𝑖√2𝑘⁵𝑡𝐸×sin(𝛼(𝑥𝑘)𝑥,₂)tanh(𝑘𝑥)1−3tanh²(𝑘𝑥)(𝑥,𝑡)=−𝑖𝐿⁻¹1𝑝𝐿𝐸^1𝑥𝑥𝐸₂+𝜂₁𝐸₀+𝜂₀𝐸₁,(𝑥,𝑡)=−√2𝑘²𝑡²×6𝑘3₂×11−5𝛼3²t+an𝛼h₂+10𝑘²sech²(𝑘𝑥)×1−3tanh₂(𝑘𝑥)+3𝛼²+𝛼+10𝑘²×3𝛼₂(𝑘𝑥)+𝛼−8𝑘²3−6𝑘²3tanh²(𝑘𝑥)𝑒^𝛼𝑖𝑥+×484√−29𝑘⁷t𝑡a²n𝑖hse₂ch²(𝑘𝑥)tanh(𝑘𝑥)(𝑘𝑥)1−3tanh²(𝑘𝑥)𝑒^𝛼𝑖𝑥−×2188−𝑘⁷3𝑡²t𝑖ansehc₂h²(𝑘𝑥)tanh(𝑘𝑥)(𝑘𝑥)²𝑒^{2𝛼𝑖𝑥}−√2𝑘²𝑡²×𝑖216𝑘₃3−24𝛼𝑘𝛼s3ec𝛼h²₂(𝑘𝑥)tanh³(𝑘𝑥)×sech₂+𝛼+13𝑘²𝜂₂(𝑘𝑥)tanh(𝑘𝑥)𝑒^𝛼𝑖𝑥,(𝑥,𝑡)=𝜂×₁(𝑥𝑘,𝑡)−6𝑡𝑘⁴sech⁵(𝑘𝑥)₄×++𝑡12(1s24e20𝑒c0ch8o³_{2𝑖𝛼𝑥}−s(h𝑘1𝑥1)(941𝑘𝑥c)os−hco(2sh𝑘𝑥()6𝑘𝑥))(𝑖𝛼cosh(𝑘𝑥)+2𝑘sinh(𝑘𝑥)).(15)

The other components of the decomposition series can be determined in a similar way; we can obtain the expression of 𝐸(𝑥,𝑡) in a Taylor series, which gives the closed form solutions as:𝐸(𝑥,𝑡)=2×√e2x𝑘p²1𝑖−𝛼3𝑥ta+nh1²(𝑘(𝑥+2𝛼𝑡))33𝛼²𝜂(𝑥,𝑡)=8𝑘₂+𝛼−10𝑘𝑡,−𝛼3−6𝑘²tanh²(𝑘(𝑥+2𝛼𝑡)).(16)

4. Numerical Description of the Solution

The Laplace decomposition method is used for finding the exact and approximate travelling-waves solutions of the Sch-KdV equation. Both the exact and approximate solutions obtained for n = 2 using LDM are plotted in Figure 1. It is evident that when compute more terms for the decomposition series the numerical results are getting much closer to the corresponding analytical solutions.

(a)

(b)

(c)

(d)

Figure 1 

The plots of results for solution of Sch-KdV equations with a fixed values of 𝛼=𝑘=0.05 and for different values of time. (a) Analytical solutions for 𝐸(𝑥,𝑡). (b) Numerical results for 𝐸2(𝑥,𝑡) by means of LDM. (c) Analytical solutions for 𝜂(𝑥,𝑡). (d) Numerical results for 𝜂2(𝑥,𝑡) by means of LDM.

5. Conclusion

The Laplace decomposition method is a powerful method which has provided an efficient potential for the solution of physical applications modeled by nonlinear differential equations. The algorithm can be used without any need to complex calculations except for simple and elementary operations.